OFFSET
1,2
COMMENTS
In other words, first differences give A002024.
Equals A010054 convolved with [1, 2, 3, ...]. - Gary W. Adamson, Mar 16 2010
LINKS
Harry J. Smith, Table of n, a(n) for n=1..1000
Gorka Zamora-López and Romain Brasselet, Sizing the length of complex networks, arXiv:1810.12825 [physics.soc-ph], 2018.
FORMULA
Let f(n) = floor(1/2 + sqrt(2*n)), then this function is S(n) = f(1) + f(2) + f(3) + ... + f(n).
a(n) is asymptotic to c*n^(3/2) with c=0.9428.... - Benoit Cloitre, Dec 18 2002
a(n) is asymptotic to c*n^(3/2) with c = (2/3)*sqrt(2) = .942809.... - Franklin T. Adams-Watters, Sep 07 2006
Set R = round(sqrt(2*n)), then a(n) = ((6*n+1)*R-R^3)/6. - Gerald Hillier, Nov 28 2008
G.f.: W(0)/(2*(1-x)^2), where W(k) = 1 + 1/( 1 - x^(k+1)/( x^(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A000330(A003056(n)) + (A003056(n) + 1) * (n - A057944(n)). This represents a closed form, because all of the constituent sequences (i.e., A003056, A000330, A057944) have a known closed form. - Peter Kagey, Jan 28 2016
G.f.: x^(7/8)*Theta_2(0,x^(1/2))/(2*(1-x)^2) where Theta_2 is a Jacobi theta function. - Robert Israel, Jan 28 2016
G.f.: (x/(1 - x)^2)*Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, May 30 2017
a(n) = n*(k+1)-k*(k+1)*(k+2)/6 where k = A003056(n) is the largest integer such that k*(k+1)/2 <= n. - Bogdan Blaga, Feb 04 2021
EXAMPLE
a(7) = 1 + 2 + 2 + 3 + 3 + 3 + 4 = 18.
MAPLE
ListTools:-PartialSums([seq(n$n, n=1..10)]); # Robert Israel, Jan 28 2016
MATHEMATICA
a[n_] := Sum[Floor[1/2 + Sqrt[2*k]], {k, 1, n}]; Array[a, 60] (* Jean-François Alcover, Jan 10 2016 *)
PROG
(PARI) f(n) = floor(1/2+sqrt(2*n))
for(n=1, 100, print1(sum(k=1, n, f(k)), ", "))
(PARI) { default(realprecision, 100); for (n=1, 1000, a=sum(k=1, n, floor(1/2 + sqrt(2*k))); write("b060432.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009
(Haskell)
a060432 n = sum $ zipWith (*) [n, n-1..1] a010054_list
-- Reinhard Zumkeller, Dec 17 2011
(Python)
from math import isqrt
def A060432(n): return (k:=(r:=isqrt(m:=n+1<<1))+int((m<<2)>(r<<2)*(r+1)+1)-1)*(k*(-k - 3) + 6*n - 2)//6 + n # Chai Wah Wu, Oct 16 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Robert A. Stump (bobess(AT)netzero.net), Apr 06 2001
EXTENSIONS
More terms from Jason Earls, Jan 08 2002
STATUS
approved